Theorem pw1m 7441

Description: A truth value which is inhabited is equal to true. This is a variation of pwntru 4289 and pwtrufal 16598. (Contributed by Jim Kingdon, 10-Jan-2026.)

Assertion

Ref	Expression
pw1m	|- ( ( A e. ~P 1o /\ E. x x e. A ) -> A = 1o )

Distinct variable group:   x, A

Proof of Theorem pw1m

Step	Hyp	Ref	Expression
1		elpwi 3661	. . . . . . . 8 |- ( A e. ~P 1o -> A C_ 1o )
2		df1o2 6595	. . . . . . . 8 |- 1o = { (/) }
3	1, 2	sseqtrdi 3275	. . . . . . 7 |- ( A e. ~P 1o -> A C_ { (/) } )
4	3	adantr 276	. . . . . 6 |- ( ( A e. ~P 1o /\ x e. A ) -> A C_ { (/) } )
5	1	sselda 3227	. . . . . . . . . 10 |- ( ( A e. ~P 1o /\ x e. A ) -> x e. 1o )
6	5, 2	eleqtrdi 2324	. . . . . . . . 9 |- ( ( A e. ~P 1o /\ x e. A ) -> x e. { (/) } )
7		elsni 3687	. . . . . . . . 9 |- ( x e. { (/) } -> x = (/) )
8	6, 7	syl 14	. . . . . . . 8 |- ( ( A e. ~P 1o /\ x e. A ) -> x = (/) )
9		simpr 110	. . . . . . . 8 |- ( ( A e. ~P 1o /\ x e. A ) -> x e. A )
10	8, 9	eqeltrrd 2309	. . . . . . 7 |- ( ( A e. ~P 1o /\ x e. A ) -> (/) e. A )
11	10	snssd 3818	. . . . . 6 |- ( ( A e. ~P 1o /\ x e. A ) -> { (/) } C_ A )
12	4, 11	eqssd 3244	. . . . 5 |- ( ( A e. ~P 1o /\ x e. A ) -> A = { (/) } )
13	12, 2	eqtr4di 2282	. . . 4 |- ( ( A e. ~P 1o /\ x e. A ) -> A = 1o )
14	13	ex 115	. . 3 |- ( A e. ~P 1o -> ( x e. A -> A = 1o ) )
15	14	exlimdv 1867	. 2 |- ( A e. ~P 1o -> ( E. x x e. A -> A = 1o ) )
16	15	imp 124	1 |- ( ( A e. ~P 1o /\ E. x x e. A ) -> A = 1o )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   E.wex 1540   e. wcel 2202   C_ wss 3200   (/)c0 3494   ~Pcpw 3652   {csn 3669   1oc1o 6574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-suc 4468  df-1o 6581

This theorem is referenced by:  pw1if  7442